Unit Cell Volume

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Unit Cell Volume
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To use space group information to build probable
models of crystalline materials, we need some
physical property data such as density and most
probably the volume of the unit cell.
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The unit cell parameters are expressed in terms
of angles between the unit cell edges (a, b, g)
while the vector relations are normally mapped on
a Cartesian system. Heres a diagram in which
general unit cell vectors are mapped to a
Cartesian system. Note how the b axis is made
coincident with Y and a axis lies in the X-Y
plane. Angles ? and ? are used to locate the c
axis with respect to X and Z.
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Now we can start to simplify.
Now r just needs to be expressed in terms of the
crystallographic unit cell parameters.
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The direction cosines of c with respect to X, Y,
and Z are
A relation for y comes from the standard
expression for the cosine of the angle(ß) between
a and c (right-hand terms go from a-axis to each
cartesian axis to c-axis)
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After some manipulation, the desired relation for
the volume is obtained.
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An alternative method to finding the volume
requires evaluating the determinant
Coesite Monoclinic a 7.135Å, b 12.372Å, c
7.173Å b 120.36
Resolve on Cartesian coordinates placing b along
Y and c along Z
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Now the expression for volume can be expressed in
determinant form.
Two things can be learned from this exercise

• The general relation for unit cell volume is
  simplified when the crystal system is other than
  triclinic
• A determinant based matrix can be used to
  calculate unit cell relations. This leads to the
  definition of the Metric Tensor.

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The Metric Tensor
Triclinic
Monoclinic (b-unique)
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Orthorhombic
Hexagonal-Trigonal
Tetragonal
Cubic
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The metric tensor has interesting properties.
Look at the determinant of the monoclinic tensor.
The determinant of the metric tensor produces the
square of the unit cell volume!